Invariant manifolds of admissible classes for semi-linear evolution equations

Abstract

Consider an evolution family U = ( U ( t , s ) ) t ⩾ s ⩾ 0 on a half-line R + and a semi-linear integral equation u ( t ) = U ( t , s ) u ( s ) + ∫ s t U ( t , ξ ) f ( ξ , u ( ξ ) ) d ξ . We prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p , q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that ( U ( t , s ) ) t ⩾ s ⩾ 0 has an exponential dichotomy and the nonlinear forcing term f ( t , x ) satisfies the non-uniform Lipschitz conditions: ‖ f ( t , x 1 ) − f ( t , x 2 ) ‖ ⩽ φ ( t ) ‖ x 1 − x 2 ‖ for φ being a real and positive function which belongs to certain classes of admissible function spaces.